HAL Id: tel-00194610 https://tel.archives-ouvertes.fr/tel-00194610 Submitted on 6 Dec 2007 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Heavy alkali-metal intercalated fullerenes under high pressure and high temperature conditions: Rb6C60 and Cs6C60 Roberta Poloni To cite this version: Roberta Poloni. Heavy alkali-metal intercalated fullerenes under high pressure and high temperature conditions: Rb6C60 and Cs6C60. Physics [physics]. Université Claude Bernard - Lyon I, 2007. English. <tel-00194610>
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HAL Id: tel-00194610https://tel.archives-ouvertes.fr/tel-00194610
Submitted on 6 Dec 2007
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.
Heavy alkali-metal intercalated fullerenes under highpressure and high temperature conditions: Rb6C60 and
Cs6C60Roberta Poloni
To cite this version:Roberta Poloni. Heavy alkali-metal intercalated fullerenes under high pressure and high temperatureconditions: Rb6C60 and Cs6C60. Physics [physics]. Université Claude Bernard - Lyon I, 2007.English. <tel-00194610>
Table 4.3: Bulk modulus B0 coefficients and its derivative B′
0 of the two systems obtained
from XRD data and ab initio calculations, of the C60 molecule in the two different sys-
tems as obtained from ab initio calculations and of the “interstitial” volume (<M-C>)3
obtained from EXAFS data analysis. The errors of the quantities obtained from ab initio
calculations correspond to the standard error of the best fit values while for the quantities
obtained experimentally also the error propagation has been considered.
a meaningful comparison cannot be carried out due to our method used for the estimation
of the bulk modulus. Nevertheless, the comparison of the compressibility between the two
systems, calculated within the same approximation, can be, in principle, fairly made.
4.6 Discussion
Important considerations arise by considering the different experimental and theoretical
bulk moduli given in Table 4.3. We should be aware that some of the compressibilities
defined here do not correspond to the thermodynamic ones, as they have a local character.
Nevertheless, their use has already been found to be extremely useful in the study of
isotropic [124] as well as in anisotropic systems [125].
By considering the geometric volume of the fullerene, VF , the total volume of the bcc
fulleride cell, VT , of the two M6C60 systems can be decomposed as:
VT = 2VF + nVi (4.2)
nVi represents the total “interstitial” volume in the bcc unit cell.
We have defined then a volume corresponding to the average distance between the alkali
metal atom and the first 22 carbon atoms neighbors as we did in EXAFS, that corresponds
to nVi. The evaluation from ab initio calculations of all the variables in eq. 4.2, i.e. VT ,
VF and Vi leads to a value of n equal to 22. By deriving and opportunely reorganizing
eq. 4.2 we can then conclude that with a 1% uncertainty, coming from the fact that we
64 Pressure induced distortion of the C60 molecule in Rb6C60 and Cs6C60
neglect the term associated to the fullerene molecule due to its low compressibility:
B0T
B0i=
V0T
V0T − 2V0F(4.3)
where the T and F labels apply again to total and fullerene respectively.
By using the right side term of eq. 4.3, we obtain a value of approximately 1.3 for
both systems. This value differs from the ones that can be obtained by evaluation of the
left side of eq. 4.3 using the experimental data in (Table 4.3) which give 2.3 for Rb6Cs60
and 1.8 for Cs6C60. This represents a difference of 78 and 41 % respectively with respect
to the result from the other part of the equation. Consequently we need to conclude
that the “interstitial” volume given by EXAFS decreases too rapidly to be considered as
representing the real interstitial volume. In addition, such volume reduction appears to
be more important in the Rb than in the Cs intercalation case.
Figure 4.10: Histograms of the ρ
distribution for the molecule in the
Rb6C60 (upper panel) and Cs6C60
systems (lower panel), for the dif-
ferent pressure values, respectively.
We report histograms for -1.8, -0.2,
2, 6.2 and 15.7 GPa and -1.2, 2,
6.8 and 14.3 GPa, for Rb6C60 and
Cs6C60, respectively, starting from
the right side of the graphs. The
histograms are reported within a
spacing between bars equal to 60%
meaning that more dispersed distri-
bution are represented with wider
columns.
In order to explain such disagreement, we consider the evolution of the geometry of
the molecule with pressure as given by our ab initio calculations. In Figure 4.10 we show
the histogram plot of the evolution with pressure of the distance of the carbon atoms from
4.6 Discussion 65
the geometric center of the molecule in Rb6C60 and Cs6C60.
While in the isolated C60 molecule with Ih symmetry all C atoms are equidistant from
the center, in the studied fullerides the fullerene molecule exhibits a bimodal distribution
of distances. This dispersion encountered in the case of the alkali intercalated fullerenes,
which characterizes a deformation with respect to the icosahedral symmetry, is further
enhanced under pressure. It corresponds to an elongation of the C60 cage along the
three Cartesian axes and confirms previous studies carried out on Rb6C60 and K6C60 [69]
mentioned in section 1.4.1 (Figure 1.13). The origin of the important distortion of the
C60 molecule in these alkali intercalated compounds can be probably found in the ionic
interaction between the fullerene and the intercalated ions.
Let us call ρ the distance of the C atoms from the center of the molecule. We consider
then the difference between ρ of each atom and the ρaverage quantity obtained as the
average between the 60 ρ values. With pressure, the distance of the 60 C atoms from
the center decreases and the C60 molecule preserves the same distribution of ρ, namely,
36 atoms at ρp > ρaverage and 24 with ρn < ρaverage. Nevertheless, the shape of such
distribution evolves differently for Rb6C60 and for Cs6C60 as shown in Figure 4.10.
We can then define a distortion parameter d as follows:
d =1
36
36∑
i=1
ρp(i) −1
24
24∑
i=1
ρn(i) (4.4)
In Figure 4.11 we plotted the fullerene distortion parameter d as a function of pressure
for both Rb6C60 and for Cs6C60. It increases with pressure in both cases and for all
pressures the distortion induced by Rb intercalation is higher of than for the Cs case.
At the higher pressure studied, i.e. around 15 GPa, d becomes 37 and 24 times higher
(for Rb6C60 and Cs6C60, respectively) than in the isolated molecule. This corresponds to
a 54% higher distortion of the C60 molecule in Rb6C60 compared to Cs6C60.
The nature of the pressure induced distortion can be understood by looking at the
upper panel of Figure 4.12 where we represent the C60 molecule in the bcc Rb6C60 struc-
ture surrounded by 24 Rb atoms placed at the tetrahedral sites at the 6 faces. In this
picture, the d parameter has been amplified by a factor 27 in order to better appreciate
the pressure induced deformation. The pressure induced distortion constitutes basically
an amplification of the one already observed at ambient conditions. In conclusion, the
shape modification of the C60 fullerene can be better interpreted as an elongation of the
molecule due to a traction force, in which the molecule is pulled along the 3 cartesian
axis through the Coulombic interaction between the negatively charged fullerene and the
alkali cations. The observed distortion is considerably smaller than that observed for the
66 Pressure induced distortion of the C60 molecule in Rb6C60 and Cs6C60
Figure 4.11: Evolution of the difference between the average value of ρp and the average
value of ρn for different pressure conditions for Rb6C60 and Cs6C60.
buckminster-fullerene in single crystal of three-dimensional (3D) polymers of undoped C60
by Yamanaka et al. [57]. They observe a cuboidal molecule at ambient pressure with the
distortion parameter, according to eq. 4.4, equal to 0.6, i.e. 37 and 57 times higher
that the distorion of the fullerene in Rb6C60 and Cs6C60, respectively, calculated by us at
around 15 GPa. This is essentially due to the formation of strong covalent bonds between
the molecules in the 3D polymerized structure.
We can now understand the apparent inconsistency that we found when comparing
the two terms of eq. 4.3. The pressure induced deformation of the C60 molecule leads to
an additional reduction of the alkali-carbon distances that needs to be added to simple
homogenous compressive effects.
In addition, our calculations show that the deformation of the fullerene in the case of
Rb intercalation is stronger than for Cs, consistent with the local compressibilities obtained
by EXAFS, where a higher “interstitial” volume compressibility is obtained for Rb6C60
(B0=13 GPa) with respect to Cs6C60 (B0=18 GPa). Moreover, although the C60 molecule
in Rb6C60 appears to suffer a stronger deformation than in Cs6C60, it is interesting that
its stiffness (B0=680 GPa) is higher than in the case of Cs intercalation (B0=530 GPa) as
obtained by ab initio calculations.
4.6 Discussion 67
In conclusion, this chapter reports a detailed study of the Rb6C60 and Cs6C60 systems
under pressure. In particular we coupled the complementary information obtained by XRD
and EXAFS with the result obtained by ab initio calculations in order to understand the
mechanisms taking place during the compression of such systems.
We have calculated and measured the compressibility of both systems and compared
this to that obtained by EXAFS for the “interstitial” volumes between molecules. Both for
Rb6C60 and Cs6C60 the EXAFS compressibilities appear to be too small to correspond to
an isotropic compression of the system. The analysis of the pressure induced deformation
of the C60 molecule via ab initio calculations allows us to understand such differences.
We infer that compression of the C60 molecule is accompanied by a shape-changing
deformation under pressure. This deformation is analogous to pulling the molecule through
the three orthogonal axis pointing towards the bcc faces containing the alkali metals.
Both experiments and calculations agree with a deformation of the fullerene molecule
x
z
y
Figure 4.12: Pressure induced distortion of C60. The distorted C60 molecule is represented
together with the 24 Rb atoms at the tetrahedral sites (upper panel). We report the
molecule at 15.7 GPa with an enhanced distortion such that d has been increased by a
factor 27. The 32 C atoms having a greater than average distance from the center, are
depicted in dark-grey color while the 24 C atoms having a smaller than average distance
from the center, are depicted in light-grey.
68 Pressure induced distortion of the C60 molecule in Rb6C60 and Cs6C60
which is more important for Rb than for Cs intercalation. The defined distortion parameter
of the fullerene, d, obtained by analysing the evolution of both structures under pressure,
is 54% higher in Rb6C60 than in Cs6C60 at around 15 GPa.
Chapter 5
High pressure stability of C60
molecules by alkali metal doping
in Cs6C60
In this chapter we study the Cs6C60 molecular crystal by Raman spectroscopy measure-
ments and by ab initio calculations, under high pressure. The Raman scattering data have
been collected from ambient pressure up to 45.5 GPa, at room temperature. We show that
the intercalation of Cesium atoms in the C60 crystalline structure allows preserving the C60
molecules up to the maximum studied pressure, i.e. more than twice the amorphization
pressure of solid C60. In addition, we observe that high resolution measurements allow
the detection of six new Raman modes and several partners in doublets in addition to the
Raman lines previously observed for the same system. The symmetry of the new observed
modes has been identified through our calculations.
5.1 Introduction
The high bulk modulus values calculated for the C60 molecule (700 and 900 GPa) [43, 44]
make the C60 fullerene a good candidate for the constitution of a molecular solid able
to sustain very high pressures, a domain which is usually reserved to simple molecular
systems, with a very limited number of atoms as diatomic molecular solids. In the pristine
fcc structure, i.e. the natural association of the C60 fullerenes, such expectations are
frustrated by the interaction between molecules, which leads to the amorphization of the
structure observed at 22 GPa at room temperature [47, 46]. From that pressure, the
signature of the molecular integrity, corresponding to Raman molecular modes, is lost,
70 High pressure stability of C60 molecules by alkali metal doping in Cs6C60
implying the destruction of the molecule.
In the present work, we show that the intercalation of solid C60 with Cs alkali atoms,
leading to the formation of the Cs6C60 compound, allows the molecules to bypass such a
limitation, warranting the stability of the C60 molecules at pressures of at least 45 GPa.
5.2 Experimental details
The method employed for the synthesis of the Cs6C60 compound is reported in appendix
A. The quality of the obtained sample was verified by XRD and is reported in chapter 4.
The sample was loaded into a gasketed diamond anvil cell in a glove box using solid
NaCl as pressure transmitting medium. The pressure was calibrated by the R1 fluorescence
line of a ruby chip placed in the vicinity of the sample. High resolution Raman spectra (∼0.5 cm−1) have been collected at room temperature. The laser beam (514.5 nm exciting
line of an Ar+ laser) was focused down to a 2 micrometer spot on the sample. The
optimum laser power was found to be 5 mW, measured directly before the high pressure
cell in order to avoid laser heating.
5.3 Raman measurements
For the isolated C60 molecule with Ih symmetry, only ten of the 46 calculated vibrational
modes are Raman active (2Ag+8Hg). For the solid fcc C60 system, the symmetry-lowering
perturbation due to the crystal field associated with the condensed phase gives rise to a
very large number of allowed Raman modes. Group-theoretical analysis performed by
Dresselhaus et al. [126] showed that the solid C60 with T6h symmetry, displays 29 one-
dimensional Ag modes, 29 two dimensional Eg modes and 87 three-dimensional Tg modes.
Nevertheless, several experiments [127, 128, 46] suggested that most of these modes are
very weak or give rise to small unresolved splittings of the ten main Raman-allowed modes
of the isolated molecule.
For the fully doped Cs6C60 compound with T5h symmetry, calculations [126] show that
each of the five-dimensional Hg modes appearing in the isolated C60 molecule splits into a
two dimensional Eg and a three dimensional Tg modes. Moreover, the 3T1g, 4T2g and 6Gg
modes in the Ih symmetry should become weakly Raman active changing their symmetry
into 3Tg, 4Tg and 6(Ag+Tg), respectively.
Low-resolution (6 cm−1) Raman measurements performed at ambient conditions for
the fully doped systems M6C60 (with M=K, Rb and Cs) have been previously reported
[128, 129, 126]. In that work, ten Raman modes corresponding to the Raman active
5.3 Raman measurements 71
modes of the isolated molecule were observed. Hence, they labelled these vibrations for
convenience with the names of the irreducible representation of the Raman active modes
of the isolated C60 molecule with Ih symmetry, i.e. 2Ag+8Hg. In addition, the authors
also observed five new lines, corresponding to partners in doublets. In particular, for the
Cs6C60 system, four of the Hg modes (Hg(2), Hg(3), Hg(5) and Hg(7)) were split into
doublets by a measurable amount. For K6C60 and Rb6C60 they also observed a splitting
of the Hg(1) mode into a doublet. A more detailed discussion about theoretical predictions
compared to experimental observation of new lines in M6C60 solids due to the symmetry
lowering from Ih to T5h can be found in Ref. [126].
In the present work, we observe six new Raman modes and eight partners in doublets
in addition to the ten Raman active modes of the isolated molecule. The behavior of all
the observed lines is followed with pressure.
In Figure 5.1 we display the Raman scattering spectra in the frequency regions 220-900
cm−1 and 1050-1680 cm−1 respectively, at various pressures at room temperature. The
behavior of all the observed lines is followed with pressure. In the following we refer to the
previously ten observed intramolecular Raman vibrations as the 2Ag and 8Hg symmetry
modes of the free molecule in order to keep the same nomenclature as used in the past.
On the other hand, we label the new observed lines according to the symmetry of the
irreducible representation of the Raman active modes of the Cs6C60 system with T5h space
group.
The Raman spectra of Cs6C60 were measured from ambient pressure up to 45.5 GPa,
at room temperature. Only part of all collected spectra are represented in Figure 5.1 for
clarity.
The six new modes are labelled in Figure 5.1 as Tg(α), Tg(β), Tg(γ), Tg(δ), Tg(ε) and
Ag(α) and their symmetry has been identified through our ab initio calculations. Many
other low-frequency Raman active modes of solid Cs6C60 associated with intermolecular
and molecule-alkali metal motions have been anticipated but they have not yet been
observed as they are too weak to be experimentally detected.
Let us first discuss our Raman spectrum at ambient pressure. The ambient pressure
frequency of all the observed Raman modes is reported in Table 5.1. The ambient pressure
data have been collected on a sample in a closed glass capillary and they show lower quality
compared to the high pressure spectra. It is then less evident to distinguish the Tg(α)
and Tg(β) modes whose evolution as a function of pressure has been clearly followed. For
these two modes, the ambient pressure Raman frequency has been extrapolated by using
the curbes fitted to the pressure evolution.
72 High pressure stability of C60 molecules by alkali metal doping in Cs6C60
Mode Mode ω0 ∂ω/∂P ∂ω/∂P
cm−1 cm−1/GPa cm−1/GPa
C60 Cs6C60 Cs6C60 C60 Cs6C60
up to 22 GPa up to 22.3 GPa
Ih Th exp. (this work) Ref. [130], Ref. [46] this work
Hg(1) Eg(1)+Tg(1) 269.9 1.1 2.0
† / 2.2
Hg(2) Eg(2)+Tg(2) 422.3 2.4, 0.16 0.3
427.6 / 0.3
T2g Tg(α) 460.3 / 0.5
T1g Tg(β) 476.0 / 1.1
Ag(1) Ag(1) 494.9 0.75, 0.94 3.0
Hg(3) Eg(3)+Tg(3) 657.5 -0.92, -0.55 -0.4
658.9 / -0.2
T2g Tg(γ) 677.1 / 0.1
Hg(4) Eg(4)+Tg(4) 755.9 -0.71, -0.50 2.3
759.0 / 2.6
T2g Tg(δ) 730.6 / -0.1
Hg(5) Eg(5)+Tg(5) 1081.9 / 2.6
1090.0 / 2.4
Gg Ag(α)+Tg(ε) 1116.0 / 3.4
1122.1 / 3.3
Hg(6) Eg(6)+Tg(6) 1229.3 / 4.3
1235.6 / 4.8
Hg(7) Eg(7)+Tg(7) 1382.0 4.12, 2.4 4.4
N.O. / †Ag(2) Ag(2) 1429.0 3.11, 1.7 4.5
Hg(8) Eg(8)+Tg(8) 1476.8 2.73, 3.7 3.4
1491.6 / 3.5
Table 5.1: Ambient pressure experimental Raman frequencies and first derivative of the
least-square fit curves of the linear pressure dependence of all the observed Raman modes
of Cs6C60. The linear fit has been considered up to 22.3 GPa. The pressure dependence
of the pristine solid C60 Raman lines (up to 22 GPa) are also reported for a comparison.
The ’†’ symbol indicates that the fit was not possible due to the scarcity of the observed
of Cs6C60 at room temperature as a function of pressure. The modes are labelled with
the names of the irreducible representation of the Raman active modes of the free C60
molecule, for convenience. A continuous evolution of the intramolecular Raman modes
under pressure is observed up to 45.5 GPa. The Raman mode intensity has been normal-
ized to the Hg(3) (1st line) intensity mode in the upper panel and to the Hg(7) intensity
mode in the lower panel. The Raman spectrum at 0.5 GPa shows the presence of a
shoulder in the Ag(2) mode coming from an impurity also found on the gasket.
74 High pressure stability of C60 molecules by alkali metal doping in Cs6C60
The Raman shifts measured for the previously observed Cs6C60 modes, are in good
agreement with those reported in the published works [128, 126, 129].
In Table 5.1 we list the symmetry of the experimentally observed Raman modes in
the alkali-metal doped C60 system and the corresponding mode symmetry of the isolated
fullerene.
In the following, the Cs6C60 Raman spectra evolution under pressure is discussed. In
Figure 5.1 we show the evolution of the Raman spectra as a function of pressure from
ambient pressure up to 45.5 GPa and in Figure 5.2 the pressure evolution of the Raman
frequencies. The continuous evolution of all the Cs6C60 Raman modes as a function of
pressure up to 45.5 GPa proves that the presence of heavy alkali metals in solid C60
contributes to increase the pressure stability region of the C60 molecules of more than
100% in comparison to pristine fcc C60 where previous studies have shown an irreversible
transition at 22 GPa accompanied by the loss of the intramolecular C60 modes [130, 46].
In Figure 5.1 we observe that most lines show an increase of their frequency with pressure
with the exception of Hg(2), Hg(3) and Tg(δ) which slightly soften linearly with pressure
and the Tg(α) and Tg(γ) modes which essentially do not show any change in frequency. A
Raman spectrum collected at a pressure of 5 GPa after pressure release from the highest
measured pressure, shows that the Raman mode evolution is reversible.
The evolution of the Raman spectra with pressure has been firstly considered in a
reduced range of pressure (up to 22.3 GPa) in order to compare our results with those
previously obtained for the non intercalated solid C60 (see Table 5.1) up to 22 GPa.
In this limited range of pressure the frequency evolution of the observed Raman modes
can be considered with good approximation a linear function of pressure. The pressure
dependence of the eight Hg lines associated to intramolecular vibrations is similar to that of
solid fcc C60 while the Raman frequency of the two Ag modes show a stronger dependence
on pressure than for pristine C60 [130, 46]. In fact, the Raman shift of the Ag(1) mode
increases as a function of pressure and its slope (3.0 cm−1 GPa−1) is approximatively
four times larger than in pristine C60 (0.75 cm−1 GPa−1 in Ref. [130] and 0.94 cm−1
GPa−1 in Ref. [46]). As Ag(1) is a nearly 100% radial mode [131], in the case of alkali
metal intercalated C60, namely Cs6C60, the presence of Coulomb interactions contributes
to rapidly increase the “breathing” mode frequency with pressure, more than in the non-
intercalated solid C60. The compression of the C60 molecules with pressure [132] coupled
to the presence of ionic interactions in the system are probably responsible for such high
value of the pressure derivative of the Ag(1) mode, in Cs6C60. The frequency increase of
the Ag(1) radial mode upon doping, was previously explained by Jishi and Dresselhaus
5.3 Raman measurements 75
[133] by considering the variation in the electric field at the sites of the negatively charged
carbon during a vibration. This variation produces an extra radial force on these atoms
responsible for the increase in the bond-stretching force constant. In addition, the effect of
pressure is to decrease the separation between the centers of the ions and the C60 sphere
causing an increase of the electrostatic interaction and finally an increase of the Raman
shift frequency of the Ag(1) vibration. This is very similar to the case of alkali-metal
intercalated systems with small dopant ions whose increase in the Ag(1) mode frequency
with respect to solid C60 has been observed to be bigger than for larger alkali metals [129].
For the pentagonal-pinch (PP) Ag(2) mode we have found a value of 4.5 cm−1 GPa−1
which is much larger than the values obtained for solid C60 [46, 130]. On the other hand,
this value is very close to that found for Cs3C60 by Fujiki et al. [134] which is 4.0 cm−1
GPa−1. We want to remind here that the Ag(2) mode, which is a nearly 100% tangential
mode, has been observed to soften after the intercalation of Cs atoms (6 cm−1 per alkali
metal) into the C60 lattice at ambient pressure [128, 129] due to an elongation of the
average C-C bond length caused by the charge transfer to the antibonding LUMO states
of the C60 molecule [4]. Then, we observe that in Cs6C60 compounds, (as also in the case
of Cs3C60) the application of pressure increases the intraball force constant of the C60
molecule, more than in the case of pristine C60.
Finally, the two Tg(3) and Eg(3) soft modes (deriving from the Hg(3) mode) in Cs6C60
are harder than in the pristine C60 system. The most intense line at 657.5 cm−1 at ambient
pressure has a pressure coefficient of -0.4 cm−1 GPa−1 which is 30 % and 57 % smaller than
the values observed for the Hg(3) mode in solid C60 [46, 130]. This could be considered
a signature of the higher stability of the fullerene molecule in Cs6C60 with respect to the
non intercalated C60.
When considering the pressure evolution of the Raman modes up to 45.5 GPa, it turns
out that their dependence as a function of pressure is no more linear for the majority of the
Raman lines (Figure 5.2). A parabolic function, which constitutes a first approximation
of an exponential-decay function, is able to reproduce the behavior of the Raman modes
with pressure. This can be considered as an indication of anharmonic effects [135]. In
Table 5.2 we report the linear and quadratic terms corresponding to the parabolic fit of
the pressure dependence of the observed Raman modes up to 45.5 GPa.
76 High pressure stability of C60 molecules by alkali metal doping in Cs6C60
5.4 Raman spectra calculation
In parallel to experiments we performed density functional ab initio simulations to calcu-
late the zone center vibrational modes of Cs6C60 under pressure.
We have used the SIESTA [120] method discussed in chapter 2. More details about
the calculations are reported in section 4.5.
We studied the frequency evolution of the Raman modes of Cs6C60 as a function of
pressure by decreasing the lattice parameter from the experimental value found at ambient
pressure and room temperature (a=11.79 A) down to 10.00 A. This corresponds to the
pressure range [-1.2; 98] GPa.
For the different volume values, we have minimized the total energy until the forces
on atoms were smaller than 0.04 eV/A [132].
The bcc unit cell contains a total of 66 atoms and sampling of the reciprocal space
was performed using a 2× 2× 2 Monkhorst-Pack mesh. We obtain the dynamical matrix
using a finite differences approach, where the atoms were displaced by ∆x=0.2A. The
Hellmann-Feynman forces were calculated for all the atoms in the system, fully building
the dynamical matrix.
We used a bond polarizability model, reported in Ref. [136] and described in appendix
B, in order to calculate the intensity of the first-order off-resonance Raman scattering. The
Raman polarizability parameters of single and double bonds in C60, used in this work are
also reported in appendix B. They have been obtained in Ref. [136] by fitting experimental
off-resonance Raman spectra of solid C60 with such model.
The predicted intensity is somewhat different from that experimentally observed. This
is due to the fact that the polarizability parameters for C atoms in Cs6C60 should radically
differ from those in the isolated C60 because of the charge transfer and also because the
single and double bond lengths change with pressure. Indeed the higher the pressure the
worse is the prediction of the Raman intensities, an indication of the degradation of the
original polarizability parameters. However, the model is a practical tool to easily identify
the Raman activity of the modes, allowing us to tightly discriminate between all the 192
modes in the spectrum.
We studied the frequency evolution of Raman active modes as obtained by ab initio
calculations together with the polarizability model. The results are reported in Figure 5.2
and Table 5.1. We report only the theoretical results obtained up to 43 GPa for a more
clear comparison with experiments.
Nevertheless, the calculations show that all Raman modes observed at low pressures
5.4 Raman spectra calculation 77
Figure 5.2: Pressure dependence of the low-frequency and high-frequency Raman modes of
Cs6C60 from ambient pressure up to 45.5 GPa and 43 GPa as obtained from experiments
(left panels) and calculations (right panels), respectively. The solid lines correspond to
least-square fits to a parabolic pressure dependence for most modes. The horizontal error
bar represents the pressure uncertainty estimated from the Lorentzian width of the ruby
R1 line; the vertical error bar are the half width at half maximum (HWHM) of each
Raman mode.
remain active even at very high pressure (98 GPa). The ten Raman modes (2Ag+8Hg)
observed experimentally and previously labelled for convenience with the names of the
irreducible representation of the Raman active modes of the isolated C60 molecule, have
been well identified in our calculations.
We observed a splitting of each Hg mode into a two-dimensional Eg mode and a three-
dimensional Tg mode, as predicted by group theory [126] and observed from our exper-
iments. Moreover, the six new observed lines, labelled as Tg(α), Tg(β), Tg(γ), Tg(δ),
Tg(ε) and Ag(α), have been well identified in this study and they all have been found to
have three-dimensional Tg symmetry except for the Ag(α) mode. The calculated first and
second derivatives of the pressure dependence of the frequencies of all the Raman modes
are reported in Table 5.2 for a comparison with the experimentally obtained values. The
fitted curves of the calculated pressure evolution frequencies reported in Table 5.2 have
78 High pressure stability of C60 molecules by alkali metal doping in Cs6C60
been considered only up to 43 GPa in order to better compare experiments and theory.
We observe a good agreement between experiments and calculations.
In conclusion, both experiments and calculations show that the molecular character of
the C60 fullerene can be maintained through alkali intercalation up to pressures at least
two times higher (around 45 GPa) than for the previously reported pure solid C60 (around
22 GPa). At the same time, as shown in chapter 3 and also in Ref. [132] a progressive
pressure-induced deformation of the molecule takes place due to the pressure enhanced
Coulombic interaction between the fullerene and the alkali metal atoms. Our study shows
that, in spite of such -slight- deformation, the molecular character of the 60-atom fullerene
is maintained. This will allow to envisage fullerene exohedral intercalation as a path to
preserve the properties of encapsulated atoms or molecules in these or larger fullerenes, at
very high pressures.
5.4 Raman spectra calculation 79
Mode ω0 ω0 ∂ω/∂P,∂2ω/∂P2 ∂ω/∂P,∂2ω/∂P2
cm−1 cm−1 cm−1/GPa,cm−2/GPa2 cm−1/GPa,cm−2/GPa2
Cs6C60 Cs6C60 Cs6C60 Cs6C60 Cs6C60
up to 45.5 GPa up to 43 GPa
Th exp. cal. exp. cal.
Eg(1)+Tg(1) 269.9 261.5 ∗ 2.4/-0.02 3.2/-0.02
† 262.6 2.6/-0.01 3.1/-0.02
Eg(2)+Tg(2) 422.3 406.7 ∗ 0.6/-0.01 0.7/ -0.01
427.6 410.8 0.6/-0.01 0.5/-0.01
Tg(α) 460.3 450.8 0.7/-0.01 0.8/-0.01
Tg(β) 476.0 452.2 1.4/-0.02 1.7/-0.02
Ag(1) 494.9 496.7 3.8/-0.04 3.7/-0.02
Eg(3)+Tg(3) 657.5 632.4 -0.2/-0.01 -0.2/-0.01
658.9 633.6 ∗ 0.1/ -0.01 -0.2/-0.01
Tg(γ) 677.1 702.4 0.1/-0.001 0.2/-0.01
Eg(4)+Tg(4) 755.9 762.0 2.7/-0.02 3.1/-0.02
759.0 765.0 ∗ 3.1/-0.02 4.1/-0.04
Tg(δ) 730.6 766.7 -0.02/-0.003 1.0/-0.01
Eg(5)+Tg(5) 1081.9 1100.6∗ 2.9/-0.02 3.6/-0.03
1090.0 1100.7 3.8/ -0.03 4.1/-0.03
Ag(α)+Tg(ε) 1116.0 1134.3 † 4.7/-0.04
1122.1 1139.2 ∗ 3.8/-0.03 4.4/ -0.03
Eg(6)+Tg(6) 1229.3 1271.7∗ 5.2/-0.08 6.6/-0.06
1235.6 1278.2 5.2/-0.04 6.9/-0.06
Eg(7)+Tg(7) 1382.0 1441.7 4.9/ -0.02 6.8/-0.05
N.O. 1450.4∗ / 7.0/-0.06
Ag(2) 1429.0 1489.0 5.2/-0.03 8.2/-0.07
Eg(8)+Tg(8) 1476.8 1522.1 4.2/-0.05 6.3/-0.05
1491.6 1528.9∗ 3.7/-0.01 5.9/-0.04
Table 5.2: Ambient pressure Raman frequencies and first and second derivatives of the
least-square fit to a parabolic pressure dependence for all modes for both experiments and
calculations. In the latter case a fit up to 43 GPa, instead of 98 GPa, has been considerend
in order to better compare the results with the experiment. The ’*’ symbols refer to the
Tg symmetry modes. The ’†’ symbol indicates that the fit was not possible due to the
scarcity of lines, observed under pressure.
80 High pressure stability of C60 molecules by alkali metal doping in Cs6C60
Chapter 6
High pressure phase transition in
Rb6C60
In this chapter we present high pressure (HP) and high temperature (HT) studies, carried
out on Rb6C60, by performing x-ray diffraction (XRD), x-ray absorption (XAS) and Ra-
man spectroscopy measurements. In particular, the occurrence of a reversible transition
has been clearly identified at around 35 GPa both at 600 K and at room temperature, by
XRD and Raman measurements, respectively. The XRD data show that the new phase
is compatible with a hexagonal unit cell structure with a=8.360(2) A and c=14.830(7) A
at 43 GPa and 600 K. At the Raman transition, Raman spectra exhibit an abrupt change
in the frequency of the normal vibrations. We speculate that the observed first order
bcc → hexagonal structure transition is accompanied by a 2D polymerization of the C60
molecules along the (001) plane.
6.1 Introduction
As discussed in more detail in chapter 1, the initial motivation of this work was the
synthesis under high temperature and high pressure conditions of three dimensional C60
polymers in the presence of alkali metals. In fact, these new materials are expected to
exhibit both high hardness and superconducting properties.
In this chapter, we present an accurate study on the evolution of the Rb6C60 system,
under high pressure and high temperature conditions, in order to check for the occurrence
of phase transitions leading to the formation of such high-dimensional polymers.
On the other hand, in the previous chapter we reported a Raman spectroscopy study
showing that the Cs6C60 compound displays an exceptional structural stability under
82 High pressure phase transition in Rb6C60
high pressure [137] (up to 45 GPa). In the present chapter, the Raman scattering study
of Rb6C60 as a function of pressure allows to compare the high pressure evolution of the
intramolecular vibrations to the case of Cs6C60.
6.2 Experimental details
The method employed for the synthesis of the Rb6C60 compound is reported in appendix
A. The XRD data reported in the next section evidence the quality of the obtained sample.
HP and HT XRD and XAS experiments were performed using diamond anvil cells
(DACs) with the x-ray beam traversing the two 2.5 mm diamonds.
The energy dispersive XAS measurements were performed at the insertion device ID24
beamline [138, 139] in transmission geometry. The beam was focused down to 10µm×10µm
FWHM (full width at half maximum).
Angular dispersive XRD experiments were performed at the insertion device ID27
beamline [110] by angle-resolved measurements. The monochromatic beam was selected
to have wavelength λ=0.3738 A and λ=0.2647 A for high temperature and room tem-
perature measurements, respectively. The focal spot size was 7µm×7µm (FWHM). The
diffraction patterns were recorded on a fast large area scanning MAR345 image plate and
ona detector-Bruker CCD for the high temperature and room temperature experiments,
respectively.
High pressure (HP) room temperature (RT) Raman spectra were recorded on a Jobin-
Yvon HR-800 Labram spectrometer with double-notch filtering and air cooled CCD de-
tector (ENS, Lyon). More experimantal details concerning the Raman experiment can be
found in section 5.2.
For all the experiments, the pressure was measured in situ before and after each mea-
surement using the R1 fluorescence emission of a ruby chip placed into the gasket hole.
We used LiF as pressure transmitting medium for HP-HT XRD measurements and NaCl
for the HP-RT XRD and Raman spectroscopy measurements. No pressure transmitting
medium was used during the XAS measurements in order to provide a more suitable
sample thickness for a good signal-to-noise ratio.
The HT conditions for XAS and XRD experiments were achieved by external resistive
heating and the temperature was measured using a thermocouple placed on the back of
one diamond.
For HP-HT XAS and XRD experiments we used a Re gasket with a 100 µm diameter
hole while for the HP-RT temperature Raman measurements we used a stainless steel
6.3 XRD measurements 83
gasket with a 100 µm diameter hole.
6.3 XRD measurements
We performed XRD measurements in the P-T range of [AP:49.5 GPa] and [RT:600 K].
We studied the evolution of the solid structure of Rb6C60 as a function of pressure both
at HT (600 K) and at RT. XRD patterns were collected after each pressure increase by ∼1.5 GPa. In Figure 6.1 we report only part of the collected patterns from 7 up to 43 GPa
at HT (lower panel) and from 6.5 up to 49.5 GPa at RT.
The data set collected at 600 K displays a clear phase transition of Rb6C60 towards a
lower-symmetry structure corresponding to the appearence of additional Bragg reflections
starting from 36.5 GPa. At 43 GPa, which is the highest measured pressure, the phase
transition was observed to be complete.
Two different experiments were performed in order to check the transition reversibility
dependence on the thermodynamic path. Firstly, we performed the pressure release at
high temperature and once we reached ambient pressure we cooled the sample down to
room temperature. The second experiment consisted in cooling the sample down to room
temperature at high pressure and then we performed the pressure release.
In both experiments the transition was observed to be reversible but with different
hysteresis ranges. The body-centered cubic phase was back at 32 GPa and at 11 GPa in
the first and the latter case, respectively. In Figure 6.2 we display the XRD data collected
at 600 K during the pressure release showing the reversibility of the transition.
For data collected at RT as a function of pressure (upper panel of Figure 6.1) it is
difficult to establish which structural modification is taking place above 45.6 GPa, due
to the broadening of the Bragg reflections at such pressure. Nevertheless, we can observe
that such increase in the Bragg peaks width is discontinuous, suggesting the occurrence
of a structural change in correspondance of such discontinuity.
All data were analyzed using the FULLPROF package [114] before and after the phase
transition within the Le Bail configuration.
The new structural phase of Rb6C60 was found to be compatible with a hexagonal
unit-cell structure by indexing the powder diffraction patterns with a dichotomy method.
In particular we used the DICVOL program [141]. Among the different solutions only the
lattice parameters of the hexagonal cell were found to be compatible with a network built
of C60 molecules. The resulting lattice parameters at 43 GPa and 600 K are a=8.360(2)
A and c=14.830(7) A. The corresponding c/a ratio is 1.774, only 8% bigger than the ideal
84 High pressure phase transition in Rb6C60
Figure 6.1: Upper panel: room temperature and high pressure XRD patterns of Rb6C60
and NaCl (used as pressure transmitting medium), normalized for acquisition time and
collected at λ=0.2647 A. The B1→B2 NaCl is also observed at around 39 GPa [140].
Lower panel: high temperature (600 K) and high pressure XRD patterns of Rb6C60 and
LiF (used as pressure transmitting medium), normalized to LiF (111) Bragg reflection
intensity and collected at λ=0.3738 A. All data are reported without any background
subtraction.
axial ratio c/a for a hexagonal close-packed crystal structure (1.633). The corresponding
Le Bail fit performed by imposing the P6/mmm space group is shown in Figure 6.3 and
6.4 XAS measurements 85
Figure 6.2: XRD data collected on Rb6C60 at 600 K, from the highest measured pressure
(43 GPa) down to 32 GPa during the pressure release. The observed XRD transition is
reversible and the cubic phase is back at 32 GPa. Data are normalized to the LiF (111)
Bragg reflection intensity.
the weighted profile R-factor which gauges the quality of the fit is Rwp=6.5% (χ2=0.29).
Unfortunately, due to the quality of the data at high pressure, a Rietveld-type refinement
was not possible, preventing us from obtaining information about the atomic positions in
the unit cell.
6.4 XAS measurements
XAS spectra of Rb6C60 were collected at the Rb K-edge (15.2 keV) as a function of
pressure both at room temperature and at 600 K. The XANES (x-ray absorption near
edge structure) signals of some of the collected spectra are shown in Figure 6.4. Due to
the unavoidable Bragg reflections from the diamond anvils, data were collected up to 200
eV above the edge, preventing EXAFS (extended x-ray absorption fine structure) analysis.
Since a detailed XANES analysis was not possible due to the difficulty in reproducing
the signal at ambient conditions, in the following we only report qualitative considerations
86 High pressure phase transition in Rb6C60
Figure 6.3: Le Bail fit of the XRD pattern collected on Rb6C60 at 43 GPa and 600 K
(Rwp=6.5 % and χ2=0.29). The ticks correspond from bottom to top to LiF and Rb6C60.
The data and the fit curve are reported in dotted and solid line, respectively. The lower
line corresponds to the residual pattern.
on the pressure evolution of the spectra. For this reason, in Figure 6.4 we report two
different views of the evolution of the XANES spectra as a function of pressure, in order
to better appreciate their change with the structural compression.
In both RT and HT experiments, we do not observe any clear modification with pres-
sure of the edge position within the precision of our measurements (± 0.4 eV) indicating
that no clear change in the electronic charge transfer from the alkali metals to molecules
can be established.
A continuous and progressive evolution of the XANES signal as a function of pressure
is observed both at room and at high temperature. This evolution is characterized by: i) a
progressive enlargement of the white line with a slight fall in intensity; ii) a modification of
the shape of the following XANES oscillation: this, initially exhibiting a doublet resonance,
gradually shows an intensity decrease of the first peak and globally shifts towards higher
energies; iii) the progressive disappearence of the second oscillation located at around
15.27 keV (at lower pressure).
6.5 Raman measurements 87
Figure 6.4: Two different views of the Rb K-edge XANES spectra of Rb6C60 as a function
of pressure at room temperature (upper panels) and at 600 K (lower panels).
For both experiments the release of pressure was performed at RT. For the experiment
performed at 600 K, during the pressure release we collected data down to 22 GPa while
for the experiment performed at RT, we collected data down to 15 GPa. In both cases we
observe a similar progressive reversibility of the signal. Figure 6.5 displays the XANES
spectra collected on Rb6C60, by decreasing pressure from 40 down to 22 GPa, after cooling
the sample down to RT. The pressure release was performed in this way in order to try to
metastabilize the HT-HP phase.
6.5 Raman measurements
Raman spectra of Rb6C60 were collected at ambient temperature as a function of pressure
from 0.3 GPa up to 40.3 GPa. Only a few spectra collected are reported in Figure 6.6 for
clarity.
For the fully doped Rb6C60 compound with T5h symmetry, calculations [126] show
that each of the five-dimensional Raman active Hg modes appearing in the isolated C60
88 High pressure phase transition in Rb6C60
Figure 6.5: XANES spectra collected at RT during the pressure release of the sample
studied at 600 K. Before releasing the pressure we cooled the sample down to RT.
molecule split into a two dimensional Eg and a three dimensional Tg modes. Moreover,
the 3T1g, 4T2g and 6Gg modes in the Ih symmetry should become weakly Raman active
changing their symmetry into 3Tg, 4Tg and 6(Ag+Tg), respectively.
Nevertheless, the Raman spectrum of Rb6C60 measured at ambient pressure and room
temperature and previously reported by Zhou et al. [129], showed that only ten main
modes were Raman active, corresponding to the ten Raman active modes of the isolated
molecule with Ih symmetry, i.e. 2 Ag and 8 Hg modes. In addition, they observed that
five of the eight Hg modes were split into doublets by a measurable amount, in particular
the Hg(1), Hg(2), Hg(3), Hg(5) and Hg(7) modes.
In the present work, the Raman spectrum collected at 0.5 GPa also shows that five
of the Hg modes are split into doublets by a measurable amount, in particular Hg(1),
Hg(2), Hg(3), Hg(5) and Hg(7) modes, in good agreement with Ref. [129]. In addition,
in the present high pressure study, we observe six completely new Raman lines and three
additional partners in doublets, corresponding to the Hg(4), Hg(6) and Hg(8) modes, as
also observed in our recent Raman study on Cs6C60. Many other low-frequency Raman
active modes of solid Cs6C60 associatedwith ball-ball and ball-alkali metal motions have
Andrea (Andy), Diana, Julio (Culio or Coolio), Montserrat (Mun), Guillaume, Jorge,
Sandra, Manu ...
I owe my heartfelt gratitude to my family ... for many reasons ...
Finally, most thanks go to Alex (naino) for supporting and encouraging me during all
the thesis period and for tolerating my numerous mood changes (in particular) at the end
of my thesis ... I liked a lot the stimulating scientific discussions about C60 and imogolite
...
Appendix A
Sample synthesis
In this appendix we explain the method that we used for the synthesis of the samples.
Even though the synthesis procedure for similar systems can be found in literature [60,
144], we did not follow exactly these methods but we rather attempted a new synthesis
scheme that arose after stimulating discussions with Prof. Laurent Duclaux.
All the preparation process has been performed in the chemistry and in the high
pressure laboratory at the LPMCN of the University Lyon 1, in Lyon. In the following we
describe the main steps:
• Fullerene cleaning: the first step consisted in the fullerite cleaning. The C60 powder
enclosed in a sealed quartz tube was heated at 550 K and pumped at the same time
at 5*10−7 mbar during ten hours.
• Stoichiometric mixing: the Rb6C60 and Cs6C60 compounds were prepared by mixing
stoichiometric amounts of annealed C60 (99.95+ % purity) with Cs and Rb (99.98
% purity) metals in glove box, respectively. In addition, a small excess amount of
alkali metals were used (respectively 2.5% and 3% wt of Cs and Rb) because they
were observed to remain on the quartz tube after the synthesis. In Figure 8.11 we
show a picture of the glove box used for avoiding sample degradation. The mixing
process must be performed in inert atmosphere in order to avoid sample reaction
or contamination due to the presence of the alkali metals, extremely sensitive to air
and humidity.
• Vacuum sealing: the mixed powder placed in a quartz tube was pumped in vacuum
at 5*10−7 mbar during ten hours (as for the fullerite cleaning) before sealing.
• Reaction: the solid state reaction was obtained by annealing the mixed powder
enclosed in a sealed evacuated quartz tube at high temperature for several days. The
134 Appendix A
Figure 8.11: Picture of the glove box employed during the sample synthesis in order to
avoid reaction or contamination of both the Rb and Cs alkali metals and the final Rb6C60
and Cs6C60 compounds.
sealed tube is placed in the middle zone of a tubular furnace, shown in Figure 8.12
where temperature is almost homogeneous. During the first year of my PhD, we
performed many different sample synthesis by varing these parameters (temperature
and time) before obtaining an excellent data quality. After each synthesis process we
verified the sample quality by XRD. The best data quality was obtained by annealing
the mixed powder at 600 K for 35-40 days.
furnace
quartz tube
Figure 8.12: The picture shows the quartz tube containing the powder sample pumped at
5*10−7 mbar. After sealing the ampoule is placed in the central zone of the furnace.
The annealing time was found to be a critical parameter for the synthesis of the
M6C60 (M=Rb and Cs) compounds. By increasing it from a few days (two or three)
up to 40 days, the result improved considerably. In particular, we observed that no
135
sign of the M6C60 compound was present in the XRD patterns for the first case (a
few days), while a high percentage of the M6C60 phase was synthesized, in addition
to other stoichiometric compounds, like M3C60 and M4C60, in the second case (40
days). Moreover, we observed that by increasing the annealing temperature from
550 up to 600 K the data quality further improved. In particular, the samples
prepared at 600 K during 40 days were pure M6C60 phase (Figure 8.13, (c) and (d)).
We also performed a synthesis of the two compounds by increasing the annealing
temperature up to 700 K and by decreasing the “reaction”time down to 15 days.
The sample characterization performed by XRD and Raman spectroscopy displayed
the lack of Bragg reflections of the M6C60 phase.
(a) (b)
(c) (d)
Figure 8.13: XRD patterns collected on the some of the different samples synthesized dur-
ing my first year of PhD. The experimental data are reported together with the simulated
patterns obtained by considering the crystalline structures reported in literature [113].
The different synthesis have been performed by varing both temperature and time in the
annealing process. For Rb6C60: (a) 550 K-3 days; (b) 600 K-30 days and (c) 600 K-40
days. For Cs6C60 (d) 600 K-40 days.
The obtained Rb6C60 and Cs6C60 powder compounds are extremely sensitive to air
136 Appendix A
and humidity. For this reason the sample preparation for the high pressure and high
temperature experiments has been performed in glove box (Figure 8.11).
In Figure 8.13 we report some XRD patterns collected on different synthesized samples
in order to check the good stoichiometry of the obtained compounds. The shown XRD
data have been collected at the ID31 beamline (ESRF) and at the BM29 beamline (ESRF).
We display four different patterns. The data shown in panels (a), (b) and (c) of Figure 8.13
are reported by following the chronological order of the corresponding synthesis. We can
observe that the quality of the sample improved considerably for the last synthesis process.
The XRD pattern reported in panel (a) corresponds to a sample obtained by annealing
the mixed powder at 550 K for three days. The Rb6C60 compound is not present in the
powder. The sample whose XRD pattern is shown in panel (b) displays better quality
compared to the previous one, the presence of both Rb6C60 (∼ 85 % in wt) and Rb4C60
(∼ 15 % in wt) compounds. The synthesis was performed by annealing the powder at
600 K for 30 days. The most recent sample synthesis, reported in panel (c) was annealed
at 600 K for 40 days and the corresponding XRD patterns shows the good quality of the
sample. The XRD pattern reported in panel (d) corresponds to the Cs6C60 compound,
synthesized using the same values of the temperature and time variables employed for the
synthesis of the Rb6C60 whose XRD pattern is shown in panel (c). The data quality is
excellent.
Appendix B
Bond polarizability model for the Raman spectra of fullerenes
In this appendix, we describe the bond polarizability model reported in Ref. [136] and
used in chapter 5 in order to select the Raman active modes of the C60 molecule among
the 192 calculated normal vibrations in the Cs6C60 system.
The intensity of the first-order off-resonance Stokes Raman scattering for a harmonic
system can be written as:
Iη′η(ω) = CωLω3S
3N∑
f=1
〈(ωf )〉 + 1
ωf
∣
∣
∣
∑
αβ
η′αηβPαβ,f
∣
∣
∣
2× δ(ω − ωf ) (8.1)
Here, C is a frequency-independent constant; ωL and ωS are the incident and scat-
tered light frequencies, respectively; ω ≡ ωL = ωS is the Raman shift; η and η′ are unit
vectors along the incident and scattered polarization directions, respectively; 〈n(ωf )〉 ≡[exp(βhωf ) − 1]−1 is the thermal average occupation number of mode f at temperature
T = (kBβ)−1 and the quantity Pαβ,f is the derivative of the electronic polarizability tensor
with respect to the normal coordinate for mode f . Hence, the calculation of the Raman
spectrum requires mode frequencies, mode eigenvectors and polarizability derivatives. In
terms of the mode eigenvectors, Pαβ,f is given by:
Pαβ,f =∑
lγ
[
∂Pαβ
∂uγ(l)
]
0
χγ(l|f) (8.2)
where [∂Pαβ/∂uγ(l)]0 are the electronic polarizability derivatives with respect to the
real-space atomic displacements uγ(l), evaluated at the system’s equilibrium configuration.
Here l=1,N labels the atomic sites, γ=1,3 labels Cartesian components and χ(f) are the
mode eigenvectors. In order to evaluate the derivatives in eq. 8.2, we assume that the
static electronic polarizability of a molecule can be expressed as a sum of individual bond
polarizabilities. In addition, we make the “zero order approximation”, meaning that the
138 Appendix B
bond polarizability parameters are functions of the bond lengths R only, i.e. α‖ ≡ α‖(R)
and α⊥ ≡ α⊥(R). This approximation neglects the dependence of a bond’s polarizabil-
ity on the motion of atoms not connected to that bond. The equilibrium-configuration
derivatives of the polarizability with respect to the atomic displacements uγ(l) are then
easily linked to derivatives with respect to the γ components of the three dimensional
bond vectors R(l, B) from atom l to neighbouring atom B. Being the displacement of
the l atom u(l) and keeping all other atoms fixed at their equilibrium positions, we have
R(l, B) = R0(l, B) − u(l), where R0(l, B) is the equilibrium bond vector from atom l to
atom B. We than obtain Pαβ,f as a sum of three contributions:
Pαβ,f = −∑
l
∑
B
[(
α′‖(B) + 2α′
⊥(B)
3
)
R0(l, B) · χ(l|f)δαβ + (8.3)
+[α′‖(B) − α′
⊥(B)][R0α(l, B)R0β(l, B) (8.4)
−1
3δαβ ]R0(l, B) · χ(l, f) +
(α‖(B) − α⊥(B)
R0(l, B)
)
(8.5)
×(
R0α(l, B)χβ(l|f) − R0β(l, B)χα(l|f) (8.6)
−2R0α(l, B)R0β(l, B)R0(l, B) · χ(l|f)])
]
(8.7)
Here, the primes denotes radial derivatives, the carets denote unit vectors and the sum
over B extends over the bonds connected to site l. The first term in eq. 8.7 represents the
change in the isotropic part of the polarizability induced by bond-stretching and the second
term represents the corresponding change in the anisotropic part of the polarizability. The
third term of eq. 8.7 corresponds to the change in the anisotropic part of the polarizability
induced by bond rotations. In C60, the Ag modes contribute to the first term and the
eight Hg modes contribute to the other two.
The sum over bonds in eq. 8.7 includes two types of bonds, single and double, so there
are six independent parameters that determine the Raman intensities.
In this thesis we have used the six polarizability parameters reported in Ref. [136].
They were obtained from the fits to the experimental off-resonance Raman spectrum of
C60 measured by Chase et al. [145] and they are listed in Table 8.4.3.
139
Bond type fit values (Ref. [136])
Single bond
(α′‖ − α′
⊥) 2.30 A2
(2α′⊥ + α′
‖) 2.30 A2
(α‖ − α⊥) 1.28 A3
Double bond
(α′‖ − α′
⊥) 2.60 A2
(2α′⊥ + α′
‖) 7.55 A2
(α‖ − α⊥) 0.32 A3
Table 8.3: Raman polarizability parameters for C60 used in this thesis work (chapter 5).
The six parameters are those reported in Ref. [136] and they were obtained by fitting the
experimental off-resonance Raman spectrum of C60 measured by Chase et al. [145].
140 Appendix B
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Fullerenes intercalees avec des metaux alcalins lourds sous
haute pression et haute temperature: Rb6C60 and Cs6C60
Dans cette these nous explorons le diagramme de phase des fullerenes intercalees avec
des metaux alcalins lourds, Rb6C60 and Cs6C60, a tres haute pression (up to 50 GPa) et
a tres haute temperature (de l’ambiante a 1500 K).
Ce travail inclue des experiences d’absorption de rayons X, de diffraction de rayons X,
de spectroscopie Raman, ainsi que des calculs DFT ab initio a haute pression.
Le couplage entre experiences et calculs permet d’observer que la presence de la forte
interaction ionique entre chaque molecule et les ions alcalins, empeche la polymerisation
des fullerenes sous pression. Dans le cas de Cs6C60, ceci a permis d’etendre le domaine
de stabilite en pression des molecules de C60 d’au moins un facteur deux par rapport
aux cristaux de C60 non-intercales. Dans le cas de Rb6C60 une transition reversible est
observee a 35 GPa.
Nous avons mis en evidence la deformation progressive de la molecule de fullerene sous
pression dans les systemes etudies. La compressibilite des deux cristaux a ete mesuree et
calculee.
DISCIPLINE
Physique
MOTS CLES
C60, fullerenes intercalees, diagramme de phase, diffraction de rayons X, absorption
de rayons X, spectroscopie Raman, calculs ab initio, haute pression.
INTITULE ET ADRESSE DE L’U.F.R. OU DU LABORATOIRE
Universite Claude Bernard - Lyon 1
Laboratoire de Physique de la Matiere Condense et Nanostructures